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Recursive construction of (J,L) (J,L) QC LDPC codes with girth 6

Mohammad Gholami, Zahra Haji Rahimi

Published Jun 1, 2016• 0 citations
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Abstract

‎In this paper‎, ‎a recursive algorithm is presented to generate some exponent matrices which correspond to Tanner graphs with girth at least 6‎. ‎For a J×L J×L exponent matrix E E‎, ‎the lower bound Q(E) Q(E) is obtained explicitly such that (J,L) (J,L) QC LDPC codes with girth at least 6 exist for any circulant permutation matrix (CPM) size m≥Q(E) m≥Q(E)‎. ‎The results show that the exponent matrices constructed with our recursive algorithm have smaller lower-bound than the ones proposed recently with girth 6‎

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Recursive construction of $(J,L)$ QC LDPC codes with girth 6

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‎In this paper‎, ‎a recursive algorithm is presented to generate some exponent matrices which correspond to Tanner graphs with girth at least 6‎. ‎For a $J times L$ exponent matrix $E$‎, ‎the lower bound $Q(E)$ is obtained explicitly such that $(J,L)$ QC LDPC codes with girth at least 6 exist for any circulant permutation matrix (CPM) size $m geq Q(E)$‎. ‎The results show that the exponent matrices constructed with our recursive algorithm have smaller lower-bound than the ones proposed recently with girth 6‎.

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‎In this paper‎, ‎a recursive algorithm is presented to generate some exponent matrices which correspond to Tanner graphs with girth at least 6‎. ‎For a J×L J×L exponent matrix E E‎, ‎the lower bound Q(E) Q(E) is obtained explicitly such that (J,L) (J,L) QC LDPC codes with girth at least 6 exist for any circulant permutation matrix (CPM) size m≥Q(E) m≥Q(E)‎. ‎The results show that the exponent matrices constructed with our recursive algorithm have smaller lower-bound than the ones proposed recently with girth 6‎
Recursive construction of (J,L)(J,L)(J,L) QC LDPC codes with girth 6
‎In this paper‎, ‎a recursive algorithm is presented to generate some exponent matrices which correspond to Tanner graphs with girth at least 6‎. ‎For a JtimesLJ times LJtimesL exponent matrix EEE‎, ‎the lower bound Q(E)Q(E)Q(E) is obtained explicitly such that (J,L)(J,L)(J,L) QC LDPC codes with girth at least 6 exist for any circulant permutation matrix (CPM) size mgeqQ(E)m geq Q(E)mgeqQ(E)‎. ‎The results show that the exponent matrices constructed with our recursive algorithm have smaller lower-bound than the ones proposed recently with girth 6‎.
This paper proposes a novel algorithm for constructing girth-8 quasi-cyclic (QC) LDPC codes with product-like (PL) exponent matrices, termed PL-QC-LDPC codes. To simplify the analysis of cycles in the Tanner graph of PL-QC-LDPC codes, we introduce the rectangular area (RA) and present an efficient semi-search RA algorithm to determine the minimum size of circulant permutation matrices (CPMs) in the exponent matrix. Moreover, search depth is employed to balance the trade-off between search complexity and optimization of the minimum CPM-size. Numerical results demonstrate that our proposed method, combined with masking method, can achieve smaller CPM-size and lower error floor than the greatest common divisor (GCD) method.
For any row weight L, several novel classes of (J = 5, L) and (J = 6, L) quasi-cyclic LDPC codes are deterministically constructed with girth eight. From these results, it is proved that (5, L) QC-LDPC codes with girth eight exist for any circulant permutation matrix (CPM) size P ≥ (2L + 3)(L - 1) + 1, and that girth-eight (6, L) QC-LDPC codes exist for any P ≥ 2(L+5)(L-1)+1. The two novel bounds remarkably improve the existing bounds of L2(L-1) + 1 and (L2+1)(L-1)+1, respectively. Moreover, for any column weight J and any row weight L, a construction for (J,L) QC-LDPC codes with girth eight is also proposed. This is the first deterministic and systematic construction which can generate girth-eight QC-LDPC codes with J ≥ 7.
Recently, some attentions have been focused on a class of low-density parity-check codes from affine permutation matrices, called APM-LDPC codes, which have some advantages rather than quasi-cyclic low-density parity-check (QC-LDPC) codes in the minimum-distance, cycle distribution and error-rate performance. In this letter, first, some novel explicit constructions for exponent matrices of conventional APM-LDPC codes with girths 6, 8, and 10 are investigated, and for each given exponent matrix E corresponding to an APM-LDPC code with girth 2g , 3 ≤ g ≤ 5, a lower bound Q = Q(E,g) is obtained, such that APM-LDPC codes with exponent matrix E have girth at least 2g, for each (prime) APM size m ≥ Q. Then, regarding to reduce the lower bound, an improved bound IQ is obtained, which is significantly better than some lower bounds reported for QC-LDPC codes with explicit constructions. Simulation results show that the constructed APM-LDPC codes outperform QC-LDPC codes with the same girth.
Recently, for each row weight KKK and column-weight JJJ, 3leJ<K3le J< K3leJ<K, several classes of (J,K)(J,K)(J,K) quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth 8 have been constructed explicitly such that their corresponding lower-bounds on the size of circulant permutation matrices (CPMs) have been considered small as possible. In this paper, for J=7J=7J=7, a class of (7,K)(7,K)(7,K) QC-LDPC codes with girth 8 is generated by an explicit method such that the lower-bounds of the constructed codes remarkably are better than the state-of-the-art bound (K−1)(K2+K)+1(K-1)(K^2+K)+1(K−1)(K2+K)+1.